Theorem dfiota2 5852

Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
dfiota2	|- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 5851	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		df-sn 4178	. . . . . 6 |- { y } = { x | x = y }
3	2	eqeq2i 2634	. . . . 5 |- ( { x | ph } = { y } <-> { x | ph } = { x | x = y } )
4		abbi 2737	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
5	3, 4	bitr4i 267	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
6	5	abbii 2739	. . 3 |- { y | { x | ph } = { y } } = { y | A. x ( ph <-> x = y ) }
7	6	unieqi 4445	. 2 |- U. { y | { x | ph } = { y } } = U. { y | A. x ( ph <-> x = y ) }
8	1, 7	eqtri 2644	1 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   {cab 2608   {csn 4177   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-sn 4178  df-uni 4437  df-iota 5851

This theorem is referenced by:  nfiota1  5853  nfiotad  5854  cbviota  5856  sb8iota  5858  iotaval  5862  iotanul  5866  fv2  6186